On focusing of shock waves

On focusing of shock waves

by

Veronica Eliasson

September 2007 Technical Reports from Royal Institute of Technology

KTH Mechanics

SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillst˚ and av Kungliga Tekniska H¨ ogskolan i Stockholm framl¨ agges till offentlig granskning f¨ or avl¨ aggande av teknologie doktorsexamen fredagen den 21 september 2007 kl 10.15 i sal F3, Kungliga Tekniska H¨ ogskolan, Valhallav¨agen 79, Stockholm.

Veronica Eliasson 2007 c

Universitetsservice US–AB, Stockholm 2007

On focusing of shock waves

Veronica Eliasson

KTH Mechanics, Royal Institute of Technology (KTH) SE-100 44 Stockholm, Sweden

Abstract

Both experimental and numerical investigations of converging shock waves have been performed. In the experiments, a shock tube was used to create and study converging shock waves of various geometrical shapes. Two methods were used to create polygonally shaped shocks. In the first method, the geometry of the outer boundary of the test section of the shock tube was varied. Four dif- ferent exchangeable shapes of the outer boundary were considered: a circle, a smooth pentagon, a heptagon, and an octagon. In the second method, an initially cylindrical shock wave was perturbed by metal cylinders placed in var- ious patterns and positions inside the test section. For three or more regularly spaced cylinders, the resulting diffracted shock fronts formed polygonal shaped patterns near the point of focus. Regular reflection was observed for the case with three cylinders and Mach refection was observed for cases with four or more cylinders. When the shock wave is close to the center of convergence, light emission is observed. An experimental investigation of the light emission was conducted and results show that the shape of the shock wave close to the center of convergence has a large influence on the amount of emitted light. It was found that a symmetrical polygonal shock front produced more light than an asymmetrical shape.

The shock wave focusing was also studied numerically using the Euler equa- tions for a gas obeying the ideal gas law with constant specific heats. Two problems were analyzed; an axisymmetric model of the shock tube used in the experiments and a cylindrical shock wave diffracted by cylinders in a two dimensional test section. The results showed good agreement with the experi- ments. The temperature field from the numerical simulations was investigated and shows that the triple points behind the shock front are hot spots that increase the temperature at the center as they arrive there.

As a practical example of shock wave focusing, converging shocks in an elec- trohydraulic lithotripter were simulated. The maximum radius of a gas bubble subjected to the pressure field obtained from the lithotripter was calculated and compared for various geometrical shapes and materials of the reflector.

Results showed that the shape had a large impact while the material did not influence the maximum radius of the gas bubble.

Descriptors: converging shock, Euler equations, imploding shock, Mach reflection, regular reflection, shock focusing, shock tube

iii

Preface

This doctoral thesis in fluid mechanics is a paper-based thesis of both experi- mental and numerical character. The thesis is divided into two parts in where the first part, starting with an introductory essay, is an overview and summary of the present contribution to the field of shock wave focusing. The second part consists of six papers. In chapter 9 of the first part in the thesis the respondent’s contribution to all papers are stated.

September 2007, Stockholm Veronica Eliasson

iv

Nothing shocks me. I’m a scientist.

Indiana Jones (1984)

v

Contents

Abstract iii

Preface iv

Chapter 1. Introduction 1

Chapter 2. Basic concepts 3

2.1. Governing equations 3

2.2. Shock tube theory 4

2.3. Shock reflection 6

2.4. Definition of stable converging shock waves 8

2.5. The schlieren technique 10

Chapter 3. Review of earlier work on shock wave focusing 12

3.1. Previous work on shock wave focusing 12

Chapter 4. Experimental setup 21

4.1. The shock tube 21

4.2. Method to shape the shock waves 23

4.3. The shock visualization 24

4.4. The light measurements 26

Chapter 5. Experimental results 28

5.1. Generation of polygonal shock waves 28

5.2. Shape of the shock wave close to the center of focusing 29 5.3. Light emission from converging shock waves in air and argon 36

5.4. Remarks 37

Chapter 6. Numerical simulations 39

6.1. Simulations in Overture 40

6.2. Problem formulation and setup 41

vii

6.3. Remarks 44 Chapter 7. An application of shock wave focusing 51 7.1. Reflection between a liquid-liquid and a liquid-solid interface 52

7.2. The maximum bubble radius 53

7.3. The Rayleigh-Plesset equation 54

7.4. Problem setup 55

7.5. Results 57

7.6. Remarks 61

Chapter 8. Conclusions and outlook 62

8.1. Experiments on shock wave focusing 62

8.2. Simulations of shock wave focusing 64

8.3. Simulations of weak shock wave focusing in a lithotriptor 65

Chapter 9. Papers and authors contributions 66

Acknowledgements 69

References 70

viii

Part I

Overview and summary

CHAPTER 1

Introduction

The title of this thesis is ’On shock wave focusing’ and this may cause you to ask yourself what is a shock wave?, where do they occur? and what is meant by shock wave focusing?

A shock wave is a thin discontinuous region in which quantities like pres- sure, temperature, and velocity make an abrupt “jump” from one state in front of the shock to another state behind the shock. Think of the difference in the two states as the difference in a stretch of a highway with free flow, compared to the same stretch when it is jammed up with cars, not moving at all. A shock wave propagates faster than the local speed of sound. Consider a fluid particle in a flow where a shock wave would pass, it would not know it before the shock arrives, because no information (except the shock itself) propagates faster than the local speed of sound. Further, shock waves are dissipative, which means that the entropy increases as the shock travels and its strength is reduced.

Shock waves are encountered many times during a normal day. Let us assume you wake up in the morning and go to the bathroom to brush your teeth. You turn on the water, and as it flows out and hits the sink, you see a roughly circular region with a very thin water layer centered around the streaming water. Further out from the center, the depth of this thin layer abruptly increases. This “jump” is an example of a shock wave. Later, you might drive your car to work. When the traffic flow is interrupted, say by red lights or traffic jams during rush hour, the vehicles slow down to a stop.

You brake as soon as the car in front of you brakes and then the car after you brakes; car after car behind you brakes to a stop. This “braking motion” that propagates behind you, against the direction of the traffic, can be viewed as a shock wave. Later, the same evening, you find yourself in the middle of a thunderstorm. Lightning streaks across the sky, followed by a loud crack and low rumblings. The noise is caused by a shock wave. The lightning produces extremely hot air which expands into the cool surrounding air faster than the speed of sound. The shock wave expands radially for about 10 m and then becomes an ordinary sound wave called thunder.

Perhaps you played outdoors with a magnifying glass when you were a child. By focusing the rays of the sun onto a piece of paper with the magnifying glass, the paper starts to smoke, then turn brown and maybe even catch fire.

1

2 1. INTRODUCTION

In our work we use the same idea, but instead of the sun, we use a shock wave and the magnifying glass is replaced by a shock tube.

In addition to the aforementioned examples, shock waves occur in many more situations, ranging from tiny bubble implosions to supernova explosions.

Shock waves have been the source of many accomplishments, from the medical treatment of shock wave lithotripsy (i.e. breaking of kidney stones) to the devastating consequences caused by explosions.

This thesis is a result of both experimental and numerical studies of con- verging shock waves. The first part of the thesis is organized as follows: the ba- sic preliminaries for shock wave focusing are discussed in chapter 2. A review of earlier work on shock wave focusing, both experimental and numerical, is given in chapter 3. The experimental setup used in the present study is explained in chapter 4 followed by a summary of the experimental results in chapter 5. The numerical simulations are discussed and the results are presented in chapter 6. Chapter 7 contains simulations on weak shock wave focusing in shock wave lithotripsy. Finally, conclusions of the present work and an outlook of the fu- ture is presented in chapter 8. The contribution of the author to the papers in section 2 of this thesis is stated in chapter 9.

Figure 1.1. Left: a traffic jam, photograph c Miha Skulj.

Upper right: lightning during a thunderstorm, photograph

Anna Tunska. Lower right: flowing water in a sink. c

CHAPTER 2

Basic concepts

2.1. Governing equations

The analysis of compressible flow is based on three fundamental equations, as discussed in Anderson (1990). They are the continuity equation, the momen- tum equation and the energy equation, presented here in integral form:

Z Z Z

V

∂

∂t ρdV + Z Z

S

ρV · dS = 0, (2.1)

Z Z Z

V

∂

∂t

 ρV

 dV +

Z Z

S

(ρV · dS)V = Z Z Z

V

ρfdV − Z Z

S

pdS, (2.2)

Z Z Z

V

∂

∂t

 ρ

 e + V ²

2



dV + Z Z

S

ρ

 e + V ²

2

 V · dS

= Z Z Z

V

˙qρdV − Z Z

S

pV · dS + Z Z Z

V

ρ(f · V)dV.

(2.3)

Here V is a fixed volume, V is the velocity vector V = (u, v, w) in the x, y and z directions, ρ is the density, S is the surface area of the volume V, p is the pressure acting on the surface S, ˙q is the heat rate added per unit mass, f represents the body forces per unit mass and e is the internal energy. The system of equations, (2.1)–(2.3), is closed with an equation of state. One of the simplest equations of state is the ideal gas law, which is valid for moderate temperatures and low pressures. It is given by

p = ρRT, (2.4)

where R is the specific gas constant and T is the temperature in kelvin. There exist a number of more intricate equations of state that model more complex situations, such as low temperature or high pressure flows where the intramolec- ular forces become important and cannot be neglected.

Because a shock wave has a width of only a few mean free paths, it can be described as a discontinuity. A shock wave is an irreversible process, and by the second law of thermodynamics, the entropy increases across the discontinuity.

3

4 2. BASIC CONCEPTS

This cannot be seen from equations (2.1)–(2.4), so an entropy relation must be added, as in Courant & Friedrichs (1948) or Zel’dovich & Raizer (1966).

2.2. Shock tube theory

Shock tubes are experimental devices used to study shock waves as well as thermodynamic and chemical properties. Usually, a shock tube consists of a long tube closed at both ends and separated into two parts by a thin membrane;

see Figure 2.1. The two parts are the high pressure part, called the driver section, and the low pressure part, called the driven section. The pressure in the low pressure part, p 1 , is usually lower than the atmospheric pressure, often on the order of a few kPa. The high pressure part, contains the highest possible pressure, p 4 , usually on the order of MPa.

To produce a shock wave, the driven section is evacuated from gas to a given pressure. Then the driver section is filled with gas. At a given pressure difference between the two sections, the membrane rapidly breaks, and the compressed gas in the high pressure part flows into the low pressure part. A shock wave travels forward through the low pressure part and a rarefaction wave, starting at the broken membrane, travels backwards through the high pressure part.

The flow conditions in the shock tube are shown in Figure 2.1. The sub- scripts in the figure indicate various regions: ‘1’ represents the undisturbed low pressure gas, ‘2’ is the region just behind the shock, and ‘3’ is the gas from the high pressure part which has passed through the rarefaction wave. Region ‘4’

indicates the high pressure gas not disturbed by the rarefaction wave and ‘5’

is the region behind the reflected shock.

Just before the membrane breaks, the pressure difference reaches its max- imum value; see Figure 2.1 (a). When the membrane breaks, a shock wave travels downstream in the low pressure part and a rarefaction wave travels up- stream in the high pressure part. The pressure and temperature distributions are shown in Figure. 2.1 (b)–(c). Next, the shock wave reflects from the rear (provided it is a closed shock tube) and returns. The reflected shock produces a very high pressure and temperature behind it; see Figure 2.1 (d)–(e). The dotted line, visible in Figure 2.1 (b)–(e), represents the contact surface between the high and low pressure gas. Across the contact surface there is no flow of gas, and the pressure and velocity are continuous.

The shock Mach number, M s , depends on the pressure ratio between the

high and low pressure part, p 4 /p 1 , the choice of gas used in the different parts

of the tube, and the respective temperatures of the gases. The relation between

the pressures p 1 and p 4 can be derived from equations (2.1) – (2.3) and is given

by equation (2.5). A derivation can be found in Liepmann & Roshko (1957).

2.2. SHOCK TUBE THEORY 5

p p p

T T

Low pressure High pressure

1 1 1

3

3 3

3

2 2

2

4 4

4 4 4

5 5 (a)

(b)

(c)

(d)

(e)

Membrane

Figure 2.1. Conditions in a shock tube; on top is the high and low pressure parts, separated by a membrane. (a) is the initial pressure distribution. (b) and (c) show the pressure and temperature distribution respectively after the membrane has broken and the shock has started to travel down stream in the low pressure part. (d) and (e) show the pressure and temperature distribution just after the shock has reflected from the rear wall.

p 4

p 1 = 2γ 1 M _s ² − (γ 1 − 1) γ 1 + 1



1 − γ 4 − 1 γ 1 + 1

a 1

a 4



M s − 1 M s

 ⁻

(2.5)

6 2. BASIC CONCEPTS

Here γ = c p /c v is the ratio between the specific heats for constant pressure and constant volume respectively, and a is the speed of sound. The subscripts denote the region in which the property is valid. Different methods can be used to create stronger shocks in a shock tube, such as heating the gas in the high pressure section, increasing the pressure difference between the high and low pressure part, and choosing a light gas in the low pressure part (so when the membrane bursts, the high pressure gas flows into a state close to vacuum).

Because a shock induces flow, meaning that the gas behind the shock prop- agates in the direction of the shock, shock tubes can sometimes be used as wind tunnels to look at aerodynamic aspects of high enthalpy flow. Using the shock tube as a wind tunnel has an advantage; it is relatively easy to create high pressures and high temperature flows that can be studied. A limitation is that the test time is short, on the order of micro seconds, because the test is run during the time when the shock has passed until either the contact surface or the reflected wave arrives.

For a more detailed explanation of shock tubes and the conditions during operation see Anderson (1990).

2.3. Shock reflection

When a shock wave interacts with a solid surface or another shock, there are several possible types of shock reflections that can occur. They can be divided into two groups: regular reflection and Mach reflection. A regular reflection consists of an incoming shock, i, and a reflected shock, r, and is the simplest configuration possible; see Figure 2.2 (a). For large angles between the flow and the solid surface, a single shock cannot turn the flow to a direction parallel to the wedge, so a three shock system is necessary. This is called a Mach reflection;

see Figure 2.2 (b). A three shock system consists of an incoming shock and a reflected shock connected to a Mach shock, m, in a point called a triple point.

Between the Mach shock and the reflected shock there is a slip line, denoted s in Figure 2.2 (b). The velocity of the gas on different sides of the slip line is in the same direction but not necessarily of the same magnitude. The flow deflection angle, θ, and the Mach number, M , behind an oblique shock are functions of the free stream Mach number, M ∞ , the gas constant, γ, and the shock angle, α, and are given by

θ(M ∞ , γ, α) = cot ⁻¹

 (γ + 1)M ∞ ²

2(M ∞ ² sin ² α − 1) − 1

 tan α



, (2.6)

and

M (M ∞ , γ, α) = s

(γ + 1) ² M ∞ ⁴ sin ² α − 4(M ∞ ² sin ² α − 1)(γM ∞ ² sin ² α + 1) [2γM ∞ ² sin ² α − (γ − 1)][(γ − 1)M ∞ ² sin ² α + 2] .

(2.7)

These equations can be found in NACA Report 1135 (1953).

2.3. SHOCK REFLECTION 7

(a) (b)

i i

m

M ∞ M

r r

θ

α

s

Figure 2.2. Shock reflection for a pseudo steady shock. (a) Regular reflection and (b) Mach reflection. M ∞ : free stream Mach number, M: Mach number after shock, i: incident shock, r: reflected shock, s: slip stream, m: Mach shock, θ: wedge angle, α: shock angle.

The type of reflection that occurs is dependent on M ∞ , γ, and θ. It is possible to determine the regions where the various types of reflections occur by computing lower and upper boundaries for these. The lower boundary, i.e.

the minimum shock angle for a flow with a free stream Mach number M ∞ , is given by the Mach wave angle, α M W :

α M W = arcsin(1/M ∞ ). (2.8)

If the incident shock is strong, (the flow behind the incident shock is subsonic), it is not possible for a reflected shock to exist and there cannot be any shock reflection. This is the criterion for the upper boundary of the reflection domain.

The boundary is defined by a subsonic flow behind the incident shock and α is obtained by solving equation (2.7) with the right hand side equal to one. The solution is called the sonic incident criterion and is given by

α S = arcsin s

γ − 3 + M ∞ ² (γ + 1) + p(γ + 1)[(M _∞ ² − 3) ² + γ(M ∞ ² + 1) ² ] 4γM ∞ ²

. (2.9) The upper boundary for regular reflections is defined as the maximum angle such that the flow turning angle of the reflected shock equals the flow turning angle of the incident shock. This criterion is called the detachment criterion and is found by computing the maximum angle α ^θ

from equation (2.6) and then solving

θ(M ∞ , γ, α) = θ(M 1D , γ, α ^θ

(M 1D , γ)). (2.10)

8 2. BASIC CONCEPTS

The solution to equation (2.10) is a fifth-degree polynomial in terms of sin ² α:

D 0 + D 1 sin ² α + D 2 sin ⁴ α + D 3 sin ⁶ α + D 4 sin ⁸ α + D 5 sin ¹⁰ α = 0, (2.11) where coefficients D 0 –D 5 are given by

D 0 = −16,

D 1 = 32M ² − 4M ⁴ − 48M ² γ − 16M ⁴ γ + 16γ ² − 16M ⁴ γ ² + 16M ² γ ³ + 4M ⁴ γ ⁴ ,

D 2 = −16M ⁴ + 4M ⁶ − M ⁸ + 104M ⁴ γ + 16M ⁶ γ − 4M ⁸ γ − 64M ² γ ² − 32M ⁴ γ ² + 8M ⁶ γ ² − 6M ⁸ γ ² − 56M ⁴ γ ³ − 16M ⁶ γ ³ − 4M ⁸ γ ³ − 12M ⁶ γ ⁴ − M ⁸ γ ⁴ , D 3 = M ⁸ − 64M ⁶ γ + 4M ⁸ γ + 96M ⁴ γ ² + 64M ⁶ γ ² +

14M ⁸ γ ² + 64M ⁶ γ ³ + 20M ⁸ γ ³ + 9M ⁸ γ ⁴ , D 4 = 8M ⁸ γ − 64M ⁶ γ ² − 32M ⁸ γ ² − 24M ⁸ γ ³ , D 5 = 16M ⁸ γ ² .

Only one root of equation (2.11) is real and bounded and gives the detachment criterion. The shock reflection domain is usually plotted for parameters (M, α) and is shown in Figure 2.3. Here, only the upper and lower boundaries for the reflection are plotted, along with the detachment criterion, above which no regular reflection can occur. However, Mach reflection can occur below the detachment criterion. As mentioned earlier, there are several possible shock reflection configurations, such as regular reflection with subsonic or supersonic downstream flow, Mach reflection with subsonic or supersonic flow downstream of the reflected shock, Mach reflection with a forward reflected shock, inverted Mach reflection and von Neumann reflection. Detailed descriptions of these configurations can be found in Mouton (2006) or Hornung (1986).

Several criteria for transition from a regular reflection (RR) to a Mach reflection (MR) exist. Three of these were proposed in von Neumann (1943) and since then many more have been suggested; see Ben-Dor (1992, 2006). The length scale concept was introduced in Hornung et al. (1979) and is the criterion that agrees best with pseudo steady flow in experimental shock tube facilities.

The ongoing research on transition conditions for RR↔MR is motivated by difficulties in matching theoretical and experimental results. One problem is the persistence of regular reflections well past the theoretical maximum limit and many publications address this problem; see Barbosa & Skews (2002).

2.4. Definition of stable converging shock waves

It is well known that a converging cylindrical shock wave is easily perturbed

from its original shape if there are any disturbances present in the flow. How-

ever, there is a certain measure of stability in these shock waves, because they

do not break up into several individual pieces. Instead, the regions with higher

2.4. DEFINITION OF STABLE CONVERGING SHOCK WAVES 9

1 1.5 2 2.5 3 3.5 4 4.5 5

0 10 20 30 40 50 60 70 80 90

α

Mach number No shock possible

No reflection

Mach wave condition Detachment condition

Sonic incident condition

Figure 2.3. Shock reflection domain for γ=1.4. Above the sonic incident condition, no reflection is possible. Regular re- flection is only possible in the region between the Mach wave and the detachment condition.

curvature travel faster than the planar parts, and the shock evolves into an- other shape. Several measures of stability were proposed by Fong & Ahlborn (1979):

(i) radial stability: ∆r/R, the perturbation radius normalized by the instan- taneous radius of the converging shock,

(ii) area (or volume) stability: ∆A/A, perturbation area (or volume) normal- ized by the instantaneous area A = πR ² (or volume V = 4πR ³ /4), (iii) form stability: where the angle, δ, between the wave normal and the

radius is measured.

The measure of stability is divided into two categories: absolute stability, in which case ∆r/R, ∆A/A, and δ tend to zero before the shock wave has focused and partial stability, where the parameters converge to a value much less than unity.

A polygonal converging shock wave is always assumed to be stable. If

a polygonal shock wave undergoes regular reflection, then its shape will be

10 2. BASIC CONCEPTS

preserved and will not change during the focusing process. Alternatively, it may undergo Mach reflection, thus changing its shape continuously as it focuses, but in a completely predictable way.

2.5. The schlieren technique

Schlieren techniques are often used when visualizing density gradients, e.g. in shock waves. The methods are rarely used for quantitative measurements of density gradients but are very useful for the qualitative understanding of the flow.

Optical methods for inhomogeneous media have been used for a long time.

In the early 1670’s Robert Hooke (1635-1703) demonstrated a simple version of what is known today as the shadowgraph method to observe the convective plume of a candle for several members of the Royal Society. Christiaan Huygens (1629-1695) invented a version of the schlieren technique to look for striae in glass blanks prior to making lenses from them. Jean Paul Marat (1743-1793) published the first shadowgram of thermal plumes from hot objects. Marat did not connect the thermal plumes with density gradients of a fluid; he in- terpreted it as a proof of an “igneous fluid”. The invention of the schlieren imaging technique is usually attributed to August Toepler (1836-1912), who named the technique after the german word for optical inhomogenities in glass:

‘Schlieren’ ¹ . He used a light source, a knife edge, and a telescope, not too different from today’s most common schlieren setups. Ernst Mach (1838-1916) confirmed in 1877, by using schlieren optics, that non-linear waves of finite strength could travel faster than the speed of sound, as earlier predicted by Riemann (1860). Since then, many gas dynamics phenomena have been visu- alized by the schlieren image technique. For a historical outlook and a detailed description of the schlieren optics method, see Settles (2001).

The speed of light, c, and the refraction index, n, will vary with the density, ρ, of the medium in which it is passing through. This means that light passing through a region of compressible flow is diffracted due to the density changes in the gas. The refraction index, n, can be written as a function of the density, ρ,

n ≡ c c 0

= 1 + β ρ ρ n

. (2.12)

Here β is a dimensionless constant, c 0 is the speed of light in vacuum, and ρ n

is the density at the standard state. The idea of the schlieren method is to cut off part of the deflected light before it reaches the registry device and thus produce darker (or brighter) regions on the photograph. If the density change takes place over a distance which is less than the wave length of the light then the optical method is sufficiently accurate.

1

‘Schlieren’ is the plural form of ‘Schliere’ and is capitalized in German but not in English.

2.5. THE SCHLIEREN TECHNIQUE 11 A schematic diagram of the schlieren method is shown in Figure (2.4). A light source is placed at (A) and the light becomes parallel after passing the lens L 1 . After passing the test section, the light is focused by the second lens L 2 . The focal plane of L 2 is where the image of the light source appears, (A’).

A schlieren edge is placed at (A’) to cut off parts of the light. Depending on how the light is intercepted, it will appear darker or brighter at the image plane of the test section. It is important to notice that the point where the light hits the focal plane of the test section does not change, it is just the amount of light that changes.

Light Source

Image of Light Source

Camera Position Image of Test Section Test Section

a d

b

j k

c

d’

a’

k’

j’

L₁ L₂

L₃ A A’

Figure 2.4. Schematic diagram of a schlieren system.

Depending on how the schlieren edge is shaped, the inhomogeneous media under consideration will appear in various ways. The most commonly used schlieren edge is a straight edge, which shows the density gradient in the flow normal to the edge. Usually, a knife edge is placed normal or parallel to the flow direction. It is possible to change the schlieren edge into other shapes to enhance various properties. For example, a dark-field edge produces bright higher-order features against a dark background. The dark-field filter can be set up by a spherical schlieren edge, e.g. a pin-head. The quality and the properties of the light source are also of importance for the quality of the final schlieren photograph. Usually, incandescent lamps, flash lamps, or lasers are used as light sources; see Vasil’ev (1971) and Settles (2001). Lasers are expensive tools and not necessarily better for schlieren imaging. The typical schlieren concept deals with a light source composed of individual rays that do not interact with any other rays. This is not true for a laser because it produces a parallel, monochromatic, and coherent light. A common problem is that schlieren systems with coherent laser light sources become schlieren- interferometers. This problem is further discussed and solutions are suggested by Oppenheim et al. (1966).

In the present study, we use schlieren optics to study shock waves in a

gas. However, there are many other applications for schlieren optics, even for

phenomena in liquids. This technique can be used to study everything from

air flows in model green houses (see Settles (2000)) to supersonic jets.

CHAPTER 3

Review of earlier work on shock wave focusing

3.1. Previous work on shock wave focusing

Guderley (1942) was first to analytically investigate the convergence of cylin- drical and spherical shock waves. Guderley derived a self similar solution for the radius of the converging shock wave as a function of time, which can be written as

R R c

=

 1 − t

t c

 α

. (3.1)

Here R is the radius of the converging shock wave, R c is the radius of the outer edge of the test section, t is the time, and t c is the time when the shock wave arrives at the center of convergence. Guderley found the self similar power law exponent for cylindrical shock waves to be α = 0.834. Since then, many more investigations of the self similar exponent have been performed, and some of the results are summarized in Table 1.

Self similar exponent

Guderley (1942) 0.834

Butler (1954) 0.835217

Stanyukovich (1960) 0.834

Welsh (1967) 0.835323

Mishkin & Fujimoto (1978) 0.828

Nakamura (1983) 0.8342, M s = 4.0 0.8345, M s = 10.0 de Neef & Nechtman ^∗ (1978) 0.835±0.003

Kleine ^∗ (1985) 0.832 + 0.028, -0.043 Takayama ^∗ (1986) 0.831 ±0.002

Table 1. Self similarity exponents for converging cylindrical shock waves. ^∗ Experiments.

The first experimental study on shock wave focusing was done by Perry &

Kantrowitz (1951). They used a horizontal shock tube with a tear-drop inset in the test section to create cylindrical shocks. They studied converging and

12

3.1. PREVIOUS WORK ON SHOCK WAVE FOCUSING 13 reflecting shocks, visualized by the schlieren technique, at two different shock Mach numbers (1.4 and 1.8). They found that creating perfect cylindrical shocks was more difficult for higher Mach numbers because the shock strength was increased. Perry & Kantrowitz suggested that this could be explained by irregular membrane opening times and bad membrane material. Also, a cylindrical obstacle was placed in the flow, and the result showed that the center of convergence was displaced toward the disturbed side of the shock wave. Another interesting observation was the presence of light in the center of the test section during the focusing process. This was interpreted to be an indicator of the presence of high temperatures, as the light was believed to be caused by ionized gas.

Sturtevant & Kulkarny (1976) performed experiments on plane shock waves which focused in a parabolic reflector mounted at the end of a shock tube.

Different shapes of parabolic reflectors were used. Results showed that weak shock waves focused with crossed and looped fronts while strong shocks did not. It was concluded that the shock strength governed the behavior during the focusing process and that nonlinear phenomena were important near the focal point.

Plane shocks diffracted by cones, a cylinder and a sphere were experimen- tally investigated by Bryson & Gross (1961). A plane shock with a Mach number of 2.82 was diffracted by a cylinder with a diameter of 1.27 cm. The diffraction was followed through about seven diameter downstream of the cylin- der and was visualized by a schlieren optics system. Results showed that when the shock wave impinged upon the cylinder, at first a regular reflection oc- curred. Between 40 ^◦ and 50 ^◦ from the forward stagnation point on the cylin- der, Mach reflection begins. As the Mach shocks collide behind the cylinder, a second Mach reflection is created. The experimental results were compared to Whitham’s theory, Whitham (1957, 1958, 1959) and showed good agreement.

3.1.1. Design of annular shock tubes

In annular shock tubes used to produce and study converging shock waves,

the shock must turn through a 90 ^◦ bend in order to reach the test section

and begin the convergence process. A simplified sketch of the annular part

of a shock tube is shown in Figure 3.1. Arrows indicate the direction of the

flow, the 90 ^◦ bend is indicated by a circle and the test section is indicated

by an oval. The design of the 90 ^◦ bend has a big influence on the resulting

flow after the shock passes it. The reflection and diffraction around 90 ^◦ bends

were investigated experimentally by Takayama (1978). To find an optimal

shape for the bend, six different bends were tested, from sharp to smooth

corners. Shock Mach numbers ranged from 1.1 – 6.0. Results for the sharp

bend showed that the transmitted shock did not stabilize before it reached the

end of the exit duct. The exit was located at X/L=8.0, where X was the

distance along the duct and L was the height of the shock tube. Also, the flow

14 3. REVIEW OF EARLIER WORK ON SHOCK WAVE FOCUSING

Figure 3.1. A simplified sketch of an annular shock tube.

The 90 ^◦ bend is indicated by a circle and the test section is indicated by an oval.

behind the shock never became uniform. Takayama stated that a sharp bend

“may be useless to obtain stable shock transmission.” A radius of curvature larger than (R o + R i )/2L, where R o and R i are the outer and inner radius of curvature of the walls, was enough for stable shock transmission. Converging- diverging bends with smooth walls were also investigated and results showed that they produced stable transmitted shocks. The best performance for a converging-diverging bend was obtained for R o = 120 mm and R i = 20 mm.

Another design consisting of several contraction corners was studied by Wu et al. (1980). Each contraction made sure that the Mach became an incident shock in the succeeding element. With this design, the flow separation and the strong pressure gradients behind the transmitted shock can be avoided. The method used by Wu et al. was found to be useful even if the Mach reflection was a double Mach reflection; see Ben-Dor (1981).

Smooth 90 ^◦ bends were investigated both experimentally and theoretically by Edwards et al. (1983). They used two bends (rectangular cross-section, 22x47.4 mm) with different radii of curvature, 75 mm and 150 mm. Incident planar shocks with Mach numbers in the range of 1.2 < M s < 2.8 were in- vestigated. A multi-spark light source together with a schlieren optics system enabled five recordings of the shock position during each run. Hence, it was possible to obtain the velocity of the shock at four times in the bend. As the shock entered the bend, it suffered Mach reflection at the outer wall, and the recovery time to a planar profile was faster for the shape with larger radius of curvature. The velocity of the shock at both the inner and the outer walls was influenced by the sharpness of the bend. The maximum velocity at the outer wall and the minimum velocity at the inner wall were both obtained for the sharper of the two bends. The results were compared to Whitham’s ray theory and showed adequate agreement of both the enhancement of the shock at the outer wall and the attenuation at the inner wall.

3.1.2. Stability of cylindrical shocks

An annular shock tube was used by Wu et al. (1981) to investigate the stability

of converging cylindrical shocks. Wu et al. perturbed the converging shock

with two kinds of artificial disturbances. The first disturbance consisted of

3.1. PREVIOUS WORK ON SHOCK WAVE FOCUSING 15 four rectangular webs upstream of the cylindrical test section. Results showed that a smooth square-shaped shock was formed, which later transformed into a sharp rectangular shape. As the shock expanded, vortex pairs were observed behind the shock front, and this was taken as a sign of the instability of the converging shock. The second type of artificial disturbance utilized a cylindrical rod with a diameter of 4.4 mm placed inside the test section at a radial distance of 2.54 cm from the center. As the shock wave was diffracted by the cylindrical rod, two pairs of Mach shocks and a reflected shock were generated. The shape of the cylindrical shock was perturbed, and it did not regain its symmetry during the focusing process. However, there was no observable shift in the focal point of the converging shock as compared to the case without artificial disturbances.

Both strong and weak converging shocks were investigated experimentally and theoretically by Neemeh & Ahmad (1986). An annular shock tube with a diameter of 152 mm was used to produce converging cylindrical shocks. Per- turbations were produced by cylindrical rods of various diameters placed in the path of the converging shock. The results showed that for a strong shock perturbed by a cylinder, the focal region of the collapse shifted toward the rod because the undisturbed part of the shock front traveled faster than the disturbed part. On the contrary, for a weak shock, the focal region was located outside the geometrical center because the disturbed part traveled faster than the undisturbed part. For both cases, the size of the focal region was depen- dent on the rod diameter. Neemeh & Ahmad defined a perturbation factor for initially strong cylindrical shocks, ǫ = ∆R/R S , where ∆R is the distance by which the perturbed part of the shock front is displaced from its undis- turbed position and R S is the instantaneous radius of the cylindrical shock wave. The perturbation factor was measured for the experimental results and then a mathematical equation was fitted to those results:

ǫ = [F (ξ)]( R S

R 0 ) ^{− G(ξ)} . (3.2)

Here ξ = d/R 0 , where d is the diameter of the cylindrical rod, R 0 is the radius at which the rod is placed, and the functions F and G are given by

F (ξ) = 0.182(ξ) − 24.59(ξ) ² + 349.19(ξ) ³ − 118.6(ξ) ⁴ , (3.3)

G(ξ) = 0.67 + 3.22(ξ) − 38.7(ξ) ² + 121.4(ξ) ³ . (3.4)

Takayama et al. (1984) used a horizontal annular shock tube to produce

converging shock waves with initial shock Mach numbers in the range of 1.10

– 2.10. The supports for the inner tube consisted of cylindrical rods with

a diameter of 12 mm and the area contraction due to the supports was less

than 7%. A double exposure holographic interferometer was used to visualize

the converging shock wave and the flow behind it. Close to the center of

convergence, the initially cylindrical shock wave became square-shaped. This

16 3. REVIEW OF EARLIER WORK ON SHOCK WAVE FOCUSING was referred to as a mode-four instability. No relation was found between the position of the four supports for the inner tube of the annular shock tube section and the square shaped shock. Artificial disturbances in the form of 12 cylindrical rods were introduced just after the 90 ^◦ corner. Close to the focal point, a square-shaped shock was again observed, and the authors concluded that an instability of mode four existed near the center. The diverging shock resumed a stable cylindrical shape.

Takayama et al. (1987) used two different horizontal annular shock tubes to investigate the stability and behavior of converging cylindrical shock waves.

One shock tube was located in the Stoßwellenlabor, RWTH Aachen, and one in the Institute for High Speed Mechanics, Tohoku University in Sendai. One of the goals was to find out if a stable converging cylindrical shock wave could be produced. The results showed that the shape of the shock wave was very sensi- tive to disturbances in the flow. Both shock tubes were equipped with supports for the inner body, and these supports caused disturbances that changed the shape of the shock wave. The Aachen shock tube had three supports, near the center of convergence, the shock wave was always triangular, showing a mode- three instability. The Sendai tube had two sets of four supports. Although the area contraction from these supports was rather small, the converging shock showed a mode-four instability as a result of their presence. To investigate the effect of disturbances, cylindrical rods were introduced upstream of the test section in the Sendai shock tube. It was found that the shock wave was signif- icantly affected by these rods during the first part of the converging process.

Later, as the shock wave reached the center of convergence, the mode-four in- stability was again observed. Takayama et al. concluded that the disturbances caused by the supports could not be suppressed by the cylindrical rods. Also, the instability, i.e. the deviation from a cylindrical shape, was found to be more significant for stronger shocks.

To minimize disturbances in the flow, a vertical shock tube with an unsup- ported inner body was used by Watanabe et al. (1995). Special care was taken to minimize possible disturbances in the shock tube to enable production of perfect cylindrical converging shock waves. The results showed that the cylin- drical shock waves tended to be more uniform than in horizontal shock tubes with supports. Still, when the shock wave reached the center of convergence, it was not perfectly cylindrical. This was believed to be caused by small changes of the area in the co-axial channel between the inner and outer body of the shock tube. To study the influence of artificial disturbances, a number of cylin- drical rods were introduced in the flow. Different numbers of rods were used, and Watanabe et al. concluded that when there was a combination of modes, the lowest mode was strongest and suppressed the other ones.

3.1.2a. Numerical simulations. A numerical study of initially weak cylindrical

converging shock waves was done by Book & L¨ ohner (1990). The authors

3.1. PREVIOUS WORK ON SHOCK WAVE FOCUSING 17 simulated the experiments of Takayama et al. (1987) and tried to verify that a mode-four instability was present in the simulations as well. A finite element scheme with triangular grid cells and adaptive mesh refinement was used for the simulations. Results showed that weak shocks with an initial Mach number of 1.1 became square-shaped close to the focal point, while shocks with an initial Mach number of 2.1 only vaguely became square-shaped. The authors explained that this was caused by a mode-four instability growing faster than other modes. It should be noted that the square is aligned with the principal directions in the grid, and it should not be ruled out that the shape is due to discrete effects.

A numerical study by Demmig & Hehmsoth (1990) also compared results to experiments by Takayama et al. (1984, 1987). The initial conditions of the numerical simulations were taken from the experiments. The initial shape of the shock front was not circular, but was given a mode-four deformation to match the experimental results. As the shock reflected, the shock front approached a circular shape. The simulation agreed well with the experimental results.

Also, converging shock waves with higher Mach numbers were studied and electron density profiles were computed. Results showed that ionization took place behind the incident shock close to the focal point and that it increased after the reflected shock had passed.

3.1.3. Polygonal shock waves

Schwendeman & Whitham (1987) used the approximate theory of geometri- cal shock dynamics by Whitham (1957) to study the behavior of converging cylindrical shocks. They showed that a regular polygon undergoing Mach re- flection will keep reconfiguring with successive intervals, i.e. transforming from an n-corner polygon to a 2n-corner polygon and then back again. Further, Schwendeman & Whitham showed that the shock Mach number for polygonal converging shock waves, subjected to Mach reflection, will increase exactly as that for a circular converging shock. They also showed that perturbed polyg- onal shock waves with smooth corners (without plane sides), first form plane sides and sharp corners. Then the shock wave oscillates between the two con- figurations until it reaches the center of convergence and starts to reflect. This behavior was later confirmed by Apazidis & Lesser (1996) and Apazidis et al.

(2002) for a smooth pentagonal converging shock wave in both experiments and in numerical simulations. The experiments were performed in a two di- mensional chamber with a smooth pentagonal shape. A diverging shock wave was produced in the center of the chamber by an exploding device, either an exploding wire or an igniting spark. The diverging shock propagated outward until it was reflected off the walls of the chamber and then started to converge.

It was observed that the shock wave assumed the same shape as the smooth

pentagonal shape of the boundary where the reflection occurred. The curved

sides became planar, but the transforming process (five corners to ten corners

18 3. REVIEW OF EARLIER WORK ON SHOCK WAVE FOCUSING and back again etc. ) was not observed due to disturbances at the center of the chamber caused by the initial explosion that created the shock. A detailed ex- planation and the results obtained in this study are given in Johansson (2000).

The focusing and reflection behavior of initially regular polygon-shaped shocks were investigated by Aki & Higashino (1990). A finite difference scheme was used to solve the two dimensional compressible Euler equations. The gas was assumed to be ideal and inviscid. The initial Mach numbers were constant along the sides of the polygon, and they ranged from 1.4 to 2.1 in a series of various tests. An equilateral triangle regular reflection was observed and the shape of the triangle was preserved during the focusing process. For polygons with more than three sides, Mach reflection and thus a reconfiguration process was observed. All of the regular polygons tested in that study focused at the geometrical center of the initial polygonal shock. As the shock started to reflect, the shock front became rounded and straight sides were not observed anymore.

A self similar solution for the focusing process of two dimensional equilat- eral triangular shock waves was investigated in Betelu & Aronson (2001). This solution shows that the corners of the triangular shock wave undergo regular reflections and preserve the triangular shape during the whole focusing process for certain values of Mach numbers and initial conditions. The energy density is bounded for this solution, which means that the Mach number will approach a constant value at the focus. This is in contrast to symmetric polygonal shocks that suffer Mach reflection at the vertices, in which case the Mach number increases as the shock approaches the focus. However, if the criteria for regu- lar reflections for the triangular shock wave are violated, then a reconfiguring process with Mach reflection takes place.

3.1.4. Three dimensional shock wave focusing

All the previously mentioned experiments have been performed for cylindrical shock waves. Production of spherical, converging, shock waves was studied by Hosseini & Takayama (2005). A test section with transparent walls and an inner diameter of 150 mm was used. The diverging shock wave was generated by small explosives in the center of the test section. The shock wave was not spherical immediately after the explosion, but as it propagated further out it quickly approached a spherical shape. Hosseini & Takayama concluded that a diverging shock wave was always stable. The diverging shock wave reflected off the wall of the test section and started to converge. The converging shock wave kept its spherical shape until it started to interact with the detonation products.

Comparisons were made with both Guderley’s similarity law and the Chester-

Chesnell-Whitham method. Both methods showed a reasonable agreement

with the experimental data. However, the methods overestimated the speed of

the shock wave, since neither of them take into account the flow ahead of the

shock wave. The shock wave in the experiments was visualized in two different

ways, both by double-exposure holographic interferometry and by a high-speed

3.1. PREVIOUS WORK ON SHOCK WAVE FOCUSING 19 video camera (100 sequential images with a frequency of 1,000,000 images/s) with the shadowgraph method. The usage of a high speed camera was a new method to visualize the entire focusing process for an individual shock wave.

Previously, only one photograph was taken for an individual shock wave. Thus it was hard to keep exactly the same experimental conditions (Mach number, pressure, etc. ) for each shock wave.

3.1.5. Imploding detonation waves

Imploding detonation waves were generated and investigated in Knystautas et al. (1969). The authors produced a two dimensional shock front in the shape of a regular polygon consisting of 30 sides, see Knystautas & Lee (1967) for further experimental details. Knystautas et al. followed the wave structure as the detonation wave converged. It was reported that Mach reflections occurring between the sides of the polygonal wave induced a smoothing effect on the shape and that it eventually became cylindrical; the detonation wave was stable. A spectroscopic analysis suggested that high temperatures, 1.89 · 10 ⁵ K, were obtained as the shock reflected from the center, and at the same time, a bright flash was generated at the center of convergence.

The stability of cylindrical imploding detonation waves was further investi- gated by Knystautas & Lee (1971). A coaxial tube was used and the detonation wave was initiated by a high-energy spark plug at the beginning of the tube.

A cylindrical implosion chamber, with a diameter of 80 mm and a thickness of 10 mm, was mounted at the rear end of the coaxial tube. The implosion wave entered the cylindrical chamber through a converging-diverging 90 ^◦ bend to minimize the attenuation effects. In this work, an artificial disturbance in the form of a cylindrical rod with a diameter of 3.2 to 9.6 mm was introduced in the implosion chamber. The authors concluded that cylindrical converging detonation waves were stable since the bright spot at the focal point appeared at the same location in every experiment. Also, for the case with a rod of diam- eter 3.2 mm placed at the rim of the implosion chamber, the disturbances on the detonation front decreased and the wave regained its cylindrical symmetry before it collapsed.

Further investigations of imploding shock waves were made by Roig &

Glass (1977) in a hemispherical chamber. A blast wave was produced at the center of the chamber. It decayed into a detonation wave and travelled out- ward and reflected from the walls, resulting in a converging detonation wave.

The measured peak temperatures were around 5,000 K. Roig & Glass (1977)

also indicated that the temperatures obtained in Knystautas et al. (1969) were

overestimated due to the invalid use of Wein’s law, and they postulated that

the actual peak temperature probably lay below 10,000 K. A spectroscopic

temperature measurement at an implosion focus in a 20 cm diameter hemi-

spherical cavity was performed by Saito & Glass (1982). They measured the

radiation intensity distributions and fitted these to black-body curves. Results

20 3. REVIEW OF EARLIER WORK ON SHOCK WAVE FOCUSING showed that temperatures were 10,000-13,000 K for imploding shock waves and 15,000-17,000 K for imploding detonation waves.

A spectroscopical study of the light emission seen during the convergence process was also performed by Matsuo et al. (1985). Light emission was ob- served as the shock wave collapsed and the full width at half maximum of the light pulse was 2.0-2.5 µs. The luminous diameter of the plasma core from the spectrograph recordings was 5 mm and the corresponding diameter observed with a camera recording was 5-8 mm. The larger value from the camera record- ing was due to the longer exposure time, which allowed the shock wave to travel a distance of up to 3 mm during the recording. Temperature measurements ranged from 13,000-34,000 K depending on the initiation energy.

Still, after more than six decades of ongoing research in the field of shock wave focusing, open questions remain and are summarized below.

• Can a stable and repeatable converging shock be created?

• How does the shape of the shock wave influence the focusing and reflec- tion behavior?

◦ What causes the light emission during the focusing process?

◦ What is the spectrum of the emitted light?

• What parameters influence the amount of light emission?

• What models should be used for numerical simulations of shock wave focusing?

Questions marked • are considered in this thesis.

CHAPTER 4

Experimental setup

The experiments were carried out at the Fluid Physics Laboratory at KTH Mechanics. The experimental setup consists of a horizontal shock tube, a light source, and a schlieren optics system. The shock tube has a test section where shock waves are focused and reflected. The process is visualized by the schlieren system with a camera. The experimental setup is shown in Figure 4.1.

1

2

3

4 5 6

Figure 4.1. Schematic overview of the experimental setup:

1. Shock tube, 2. Pulse laser, 3. Schlieren optics, 4. PCO CCD camera, 5. Lens, 6. Schlieren edge.

4.1. The shock tube

The shock tube used in the present experimental studies is a typical setup for analysis of converging shocks. Similar setups have been used by several other investigators, see for example Perry & Kantrowitz (1951); Takayama et al.

(1984); Neemeh & Ahmad (1986). The new feature with this shock tube is that the outer boundary of the test section is exchangeable and various geometrical shapes can be used.

The 2.4 m long circular shock tube consists a high pressure part and a low pressure channel which are separated by a 0.5 mm thick aluminum membrane.

An illustration of the shock tube and its main elements is shown in Figure 4.2.

To create a shock wave, the low pressure channel is evacuated of gas to a given pressure. Then the high pressure part is filled with gas, and at a given pres- sure difference between the two parts, the membrane bursts. The shock wave becomes planar as it travels downstream in the inlet section of the low pressure

21

22 4. EXPERIMENTAL SETUP

channel. The pressures in the high and low pressure parts are monitored by sensors, see Figure 4.2.

To control the membrane opening, a knife-cross is placed in the inlet of the low pressure channel. The knife-cross helps the membrane to open evenly, shortens the time until a fully developed shock has formed, and prevents un- necessary disturbances. It helps prevent pieces of the membrane from coming loose.

1 2 3 4

6 7 8 9

5

Figure 4.2. Schematic overview of the shock tube setup: 1.

High pressure part, 2. Low pressure channel: inlet section, 3.

Low pressure channel: transformation section, 4. Low pressure channel: test section, 5. High pressure sensor, 6. Low pressure sensor, 7. Vacuum valve, 8. Vacuum pump, 9. Shock sensors.

When the plane shock wave reaches the transformation section, the shock wave is forced to become annular by a conically diverging section where the diameter increases from 80 mm to 160 mm; see Figure 4.3. The cross-sectional area is held constant from the inlet section through the transformation section.

The annular section is formed by an inner body mounted coaxially inside the wider diameter outer tube.

The 490 mm long inner body, with a diameter of 140 mm, is held in place by two sets of four supports. The two sets are placed 30.75 cm apart, and the supports are shaped as wing profiles to minimize the disturbances on the flow.

Also, the second set of supports is rotated 45 ^◦ relative to the first set. The shock speed, U s , is measured by sensors placed in the annular section. The sensors are triggered by the temperature jump caused by the passage of the shock wave.

The test section is mounted at the end of the annular part of the shock

tube. After a sharp 90 ^◦ bend, the annular shock wave enters the test section

and the focusing and reflection process begins. The gap between the two facing

glass windows in the test section is 5 mm, reducing the cross-sectional area to

half of that in the annular part.

4.2. METHOD TO SHAPE THE SHOCK WAVES 23

1

2 3

4 5 6

Figure 4.3. The annular part of the shock tube: 1. Inner body with a cone, 2. Supports, 3. Mirror, 4. Lens, 5. Glass windows for visualization, 6. Test section.

4.2. Method to shape the shock waves

Two different methods are used to create a polygonally shaped converging shock wave. In the first method, the shape of the shock wave is determined by the shape of the outer boundary of the test section. Four different outer boundaries of the test section are been used in the present experiments: a circle, a smooth pentagon, a heptagon and an octagon. The radius for the circular reflector boundary is 80 mm. The shape for the smooth pentagonal boundary is given by the following equation:

r = r 0

1 + ε cos(5θ) (4.1)

where r is the radius, ε = 0.035 and r 0 = 77 mm. The radius for the circum- scribed circle is 80 mm, both for the heptagonal and the octagonal reflector boundaries. The four reflector boundaries are shown in Figure 4.4.

(a) Circular. (b) Pentagonal. (c) Heptagonal. (d) Octagonal.

Figure 4.4. The four outer boundaries for the test section

used in the experiments.

24 4. EXPERIMENTAL SETUP

In the second method, an initially cylindrical shock wave produced by the circular outer boundary is perturbed by metal cylinders placed in various patterns and positions between the two facing glass windows inside the test section. The cylinders have three different diameters: 7.5, 10 or 15 mm. The cylinders are equipped with rubber rings on one end and glue on the other end and are held in place by the pressure between the two facing glass windows.

The method to place these cylindrical obstacles in the test section is both safe and adjustable. An example with 16 cylinders (two octagonal patterns of radius r = 46 mm) where the cylinders alternate between diameters of 10 mm and 15 mm is shown in Figure 4.5.

Annular channel 2x8 cylinders

Figure 4.5. Rear part of the shock tube with 2x8 cylinders, where every other cylinder has 15 mm and 10 mm diameter, placed in the test section at r = 46 mm.

It is worth noting that the supports were adjusted to produce a minimal disturbance for the experiments with the heptagonal reflector boundary; for the other shapes, two of the supports were not optimally positioned. The optimal position is where the chord of the wing profile is aligned with the flow, so that the wing profile is an aerodynamic body. The case with a not optimal position was obtained when the wing profile was placed so that the chord of the profile was perpendicular to the direction of the flow, thus creating a bluff body.

4.3. The shock visualization

The facing surfaces in the test section consist of glass windows, and the con- vergence and reflecting process is visualized by the schlieren optics method.

An air-cooled Nd:Yag (NewWave Orion) laser is used as a light source. The

laser is operated in single shot mode with 5 ns light pulses. The laser is placed

4.3. THE SHOCK VISUALIZATION 25 outside the shock tube, either parallel or normal to the axis of the shock tube.

If the laser is placed parallel to the axis, then a mirror is used to deflect the light through the laser light entrance on the shock tube.

The laser light entrance is a hole with a diameter of 6 mm through one of the upstream positioned supports for the inner body. When the laser light beam has entered the shock tube, it is deflected in the axial direction by a mirror placed inside the inner body. To simplify the adjustments of the optical setup, an electric motor is attached to the mirror in order to fine tune it after the equipment is in place. After hitting the mirror, the laser light enters a beam expander that produces a parallel light. The beam expander consists of two lenses. The first lens is biconcave with a diameter of 6 mm and a focal length of –8 mm. The second lens is plane convex with a diameter of 95 mm and a focal length of +300 mm. After the beam expander, the parallel light passes the first glass window, enters the test section, and then leaves the shock tube via the rear end glass window to enter the schlieren optics system.

4.3.1. The schlieren optics

The receiver part of the schlieren optics system is placed 1150 mm from the rear glass window of the shock tube. The receiver system consists of a large lens, 185 mm in diameter, with a focal length of 1310 mm and two mirrors that deflect the light into the section located at the top of the system.

The schlieren edge is placed in the image plane of the light source to cut off parts of the deflected light beams. Usually, the schlieren edge is a straight edge, but in this experiment, a spherical needle-point with a radius of 1 mm was used. This schlieren edge was chosen to match the shape of the shock wave, and it also produces a so called dark-field filter, which gives bright higher-order features against a dark background; see Settles (2001).

After passing the schlieren edge, the light goes through a lens and then enters the camera. The camera is a CCD PCO SensiCam (12 bits, 1280 x 1024 pixels, pixel size: 6.7 x 6.7 µm) equipped with a Canon lens with a focal length of 80 mm.

For experiments with the heptagonal reflector boundary and the cylindrical obstacles, special care was taken to avoid spurious light reflections inside the inner body by adding a light-absorbing coating to the interior of the inner body.

This was done to obtain higher quality photographs.

4.3.2. The shock speed measuring device and time control

Two units, each containing a sensor and amplifier, are placed in the wall of

the outer tube in the annular part of the shock tube. The sensor element is a

70 mm long glass plug with a diameter of 17 mm and a thin strip of platinum

paint at the end. The glass plug is mounted in a hole so that the end surface

(with the platinum paint) is flush with the inner surface of the tube.

26 4. EXPERIMENTAL SETUP

The resistance of platinum is temperature dependent, so when the shock wave passes the sensor, the resistance of the platinum is changed due to the temperature increase caused by the shock wave. This change in resistance is transformed to a voltage pulse via an electric circuit which can be monitored on an oscilloscope. The electric circuit consists of an amplifier, an AD845 operational amplifier with a settling time of 350 ns to 0.01%. The glass plug is shown in Figure 4.6 and the circuit diagram of the electric circuit is shown in Figure 4.7.

Platinum paint

Figure 4.6. A sensor for shock speed measurement consisting of a glass plug with a thin strip of platinum paint at the end surface.

A time delay unit (Stanford Research System, DG535) is used to control the laser and the camera to enable exposures of the converging shock wave at predetermined time instants. This is necessary because it is not possible to take more than one photograph during each run.

4.4. The light measurements

During the light emission measurements, a photomultiplier (PM) tube is con-

nected to the rear end of the shock tube. The PM tube (RCA 4526) is a light

detector, and the time resolved output signal is proportional to the number

of photons detected at each moment. The PM tube is placed in a light-sealed

plastic cover to ensure that the detected light is originating from the converging

shock wave and not from light sources within the laboratory. It is possible to

mount the PM-tube in two different positions inside the cover, and one of these

4.4. THE LIGHT MEASUREMENTS 27

OUT IN

+12 V

-12 V

100pF 6800pF

2.2

2.2 2.2

2.2 0.1 0.1 0.1 0.1

500 100 100

3.3 1.5kΩ 3.3

Figure 4.7. Circuit diagram for the amplifier. Resistances in kΩ and capacitances in µF.

will allow the use of schlieren optics and the PM-tube simultaneously. The PM

tube is operated at -1,100 V during all experiments.

CHAPTER 5

Experimental results

In this chapter, the results from the experiments on shock wave focusing are presented briefly. The goals of the experimental study were to find out if we could create stable and repeatable converging shocks and to analyze how the shape of the shock wave influenced the focusing behavior. Furthermore, we continued to investigate the light emission that appeared during the focusing process. The questions we investigated were i) “When does the light appear?”, ii) “What parameters are important for the amount of light emitted?”, and iii) “Why is there light?”

The experimental results are divided into two sections. First, results from the two methods to generate polygonal shock waves are discussed. In the second section, the results from the light emission experiments are evaluated. Papers 1–5 enclosed in the second part of this thesis provide more details and further discussions on the results presented in this chapter.

5.1. Generation of polygonal shock waves

Two different methods were used to create polygonal converging shock waves.

The first method consisted of changing the geometrical shape of the outer boundary of the test section. In the second method, an initially cylindrical shock wave was perturbed by cylindrical obstacles placed in various patterns and positions inside the test section.

Figure 5.1 shows schlieren photographs of a converging shock wave, shaped by the heptagonal outer boundary in Figure 4.4 (c), are shown. At first, the converging shock assumes the shape of the boundary, a heptagon, as in Fig- ure 5.1 (a). Next, the heptagon-shaped shock will transform into a double heptagon because the corners consists of Mach shocks that propagate faster than the adjacent planar sides; this is shown in Figure 5.1 (b). The double heptagon transforms back to a heptagon when the faster moving Mach shocks consume the planar sides. At this configuration, the resulting heptagon is ori- ented opposite to the original one, see Figure 5.1 (c). This reconfiguration process will, according to theoretical and numerical results, continue during the rest of the convergence process if there are no disturbances present. Fi- nally, the shock converges, reflects in the center, and starts to diverge. At first, the reflected shock assumes a circular shape, but it will later be influenced by

28